The 5th GTSS
GEOMETRYTOPOLOGY SUMMER SCHOOL
İstanbul Center for Mathematical Sciences
July 618, 2020
First Week
Second Week
Scientific Commitee
Vicente Cortés 
University of Hamburg, Germany

İzzet Coşkun 
University of Illinois at Chicago, USA

Ljudmila Kamenova 
Stony Brook University, USA

Lei Ni 
University of California at San Diego, USA

Tommaso Pacini 
University of Torino, Italy 
Gregory Sankaran 
University of Bath, UK 
Misha Verbitsky 
IMPA, Brasil

Organizing Commitee
Craig van Coevering 
Bosphorus University

İlhan İkeda 
Bosphorus University

Mustafa Kalafat 
Nesin Mathematical Village

RegisterTR
Poster
Participants
Arrival
Information
The 5th GTSS will be held at the
IMBM,
İstanbul Center for Mathematical Sciences
which is established in the main (South) Campus of Boğaziçi University (Bosphorus).
The first photo on top of this webpage demonstrates the view from the venue.
There will be about 15 minicourses of introductory nature, related to the GeometryTopology research subjects.
In the middle of the week there is an excursion on the Bosphorus.
Do not forget to get your swimsuit with yourself.
Application
Graduate students, recent Ph.D.s and underrepresented minorities are
especially encouraged to the summer school.
Daily expenses including bed, breakfast, lunch, dinner is around 20€.
You may stay at cheap
hostels around Taksim(Nightlife) square and take the
subway to the Campus easily.
We are trying to arrange accomodation on Campus as well.
Please fill out the application form to attend to the summer school.
Airport: İstanbul Airport  IST is the closest one. Take a
bus from the airport to the 4. Levent. Then take a taxi or subway to reach to the main/south campus.
Alternatively you can use Sabiha Gökçen Airport  SAW and take a
bus from there to 4. Levent. Then take a taxi or subway to reach to the main/south campus.
Navigate the link above for more detailed arrival and venue information.
Visas: Check whether you need a visa beforehand.
Abstracts
The Calabi Conjectures
We shall state, prove, and study applications of some of Calabi’s Conjectures.
Topics to be covered are as follows:
Definition of complex manifolds and holomorphic maps, Examples.
Kahler metrics and examples.
Holomorphic line bundles, the canonical bundle and the first Chern class.
Statement of Calabi’s volume conjecture, the representability of the first Chern class conjecture, and the KahlerEinstein conjecture.
Examples and applications.
If time permits, a gentle introduction to the ideas behind the proof (method of continuity, a priori estimates).
Textbook and/or References: 
Prerequisites:
Manifold theory including Riemannian metrics and the LeviCivita connection, complex analysis, and functional analysis. Some exposure to elliptic PDE and complex geometry will be immensely helpful but is not necessary.
Harmonic maps, constant mean curvature surfaces and integrable systems
In this lecture, we give an overview of homogeneous spaces, harmonic maps, integrable systems, constant mean curvature surfaces and their relations. First, we see how harmonic maps from a surface into a homogeneous space can be understood as integrable systems and they are related to constant mean curvature surfaces in space forms via Gauss maps (RuhVilms theorem). Finally, we give a construction method of harmonic maps (and constant mean curvature surfaces) via integrable systems (the socalled DPW method).
Daily description is as follows.
1. Overview
2. Manifolds and homogeneous spaces
3. Integrable systems and harmonic maps
4. Constant mean curvature surfaces, Gauss maps and RuhVilms theorem
5. Construction of harmonic maps via integrable systems
6. Advanced topics
Textbook, References or/and course webpage:
1.Shoichi Fujimori, Shimpei Kobayashi, Wayne Rossman, Loop Group Methods for Constant Mean Curvature Surfaces, Rokko Lectures in Mathematics, 17 (2005), v+118 pp. arXiv:math/0602570
2. Josef F. Dorfmeister, Franz Pedit, Hongyou Wu, Weierstrass type representation of harmonic maps into symmetric spaces. Comm. Anal. Geom. 6 (1998), no. 4, 633–668.
Prerequisites: Linear Algebra, Calculus
Level (erase some): Graduate Advanced undergraduate
Bers embedding of Teichmüller spaces
In these lectures I will introduce Teichmüller spaces from the point of view of AhlforsBers theory, and describe how it acquires a complex structure via an embedding into the vector space of holomorphic quadratic differentials. Along the way, I shall talk about quasiconformal maps and their boundary, the Schwarzian derivative, amongst other basic topics. The image of the Bers embedding is still mysterious, and I plan to describe some open problems.
Textbook, References or/and course webpage:
1. Teichmüller theory and applications, Volume 1, by John H. Hubbard. Published by Matrix editions, 2006.
2. Lectures on Quasiconformal Mappings, 2nd edition, by Lars V. Ahlfors. AMS University Lectures Series, Volume 38. (original 1966, reprinted 2006)
3. Univalent Functions and Teichmüller spaces, by Olli Lehto. GTM series, Springer, 1987.
Prerequisites: Complex Analysis, some knowledge of hyperbolic geometry and Riemann surfaces would be preferable.
Level (erase some): Graduate Advanced undergraduate
Topics in Stability and Fano Varieties/del Pezzo Surfaces
Monodromy and Periods of Algebraic Varieties
We discuss various topics pertaining to the topology of complex algebraic varieties, relying on classical techniques in complex algebraic geometry and algebraic topology. We will begin by introducing the Milnor fibration and construct the monodromy action on the homology of the fibers of the Milnor fibration. We will discuss vanishing cycles in this concrete setting. We will then proceed to discuss applications of these concepts in classical algebraic geometry, such as the theory of Lefschetz pencils.
In the second part of the week, we will focus on the cohomological aspects of our subject. The concept of the GaussManin connection will be introduced. For motivational purposes, we will discuss the classical theory of periods, PicardFuchs equations and similar topics in the context of the Legendre family of elliptic curves. A basic review of local systems and vector bundles will be given. Time permitting, Hodge Structures and more advanced topics may be introduced.
If the need arises, tools from basic differential topology and homological algebra may also be discussed.
Textbook or/and course webpage:
1. Singular Points of Complex Hypersurfaces, J. Milnor, 1968
2.GriffithsHarris, Principles of Algebraic Geometry
3. Otto Forster, Lectures on Riemann Surfaces, 1981
4. A Scrapbook of Complex Curve Theory, H. Clemens, 1980
5. Complex Algebraic Geometry and Hodge Theory Claire Voisin, 2002
6. Pierre Deligne, SGA 7, vol II
Advanced Topics
5. Mixed Hodge Structures, C.Peters, J.H.M. Steenbrink, 2008
6. Sheaves in Topology A. Dimca, 2004
Also see the papers:
The Topology of Complex Projective Varieties after S. Lefschetz, K.Lamotke, 1979
P.A. Griffiths, Periods of integrals on algebraic Manifolds 1970
Prerequisites:
Complex Analysis, Algebraic Topology, Intro to Algebraic Curves (preferred)
Complex Potential Theory
The aim is to cover as much as the first five chapters of the textbook.
1. Harmonic Functions
2. Subharmonic Functions
3. Potential Theory
4. The Dirichlet Problem
5. Capacity
Level: Graduate, advanced undergraduate
Textbook or/and course webpage:
Potential Theory in the Complex Plane, T. Ransford
Prerequisites:
Complex Analysis
Special metrics in Sasakian geometry
This course will begin with an introduction to Sasakian geometry, a type of metric contact structure which is an odd dimensional analogue of a Kähler structure on a complex manifold. We will then consider the problem of finding a special Sasakian metrics, such as Einstein, and more generally constant scalar curvature and extremal metrics.
Textbook or/and course webpage:
1. Boyer, Charles P.; Galicki, Krzysztof. Sasakian geometry.
Oxford Mathematical Monographs.
Oxford University Press, Oxford, 2008. xii+613 pp.
2. Futaki, Akito; Ono, Hajime; Wang, Guofang
Transverse Kähler geometry of Sasaki manifolds and toric SasakiEinstein manifolds.
J. Differential Geom. 83 (2009), no. 3, 585–635.
3. Collins, Tristan C.; Székelyhidi, Gábor. Ksemistability for irregular Sasakian manifolds. J. Differential Geom. 109 (2018), no. 1, 81–109.
Prerequisites:
A first year graduated course in differential geometry should be sufficient.
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Representation Theory of the Lie Algebra of G_2
In this lecture series we give an introduction to the representation theory of the Lie algebra of G_2.
We review Lie groups, Lie algebras, Cartan subalgebra, Root systems. Then, starting from concrete examples we will work on sl(2,C), sl(3,C), sl(4,C) and finally sl(n,C) and their representations. We also go into their geometric interpretations. Also spend time on sp(2n,C) and so(m,C) if time permits.
Textbook or/and course webpage:
1. Fulton, William; Harris, Joe  Representation theory. A first course.
Graduate Texts in Mathematics, 129. Readings in Mathematics. SpringerVerlag, New York, 1991. xvi+551 pp. ISBN: 0387975276
Lectures 11,12,13,14 and 22.
Prerequisites:
Lie groups and Lie algebras.
Minimal Submanifolds, Mean Curvature Flow and Isotopy Problems
Many fundamental results in geometry and topology have been established through the development of minimal submanifold theory and geometric flow techniques. In this mini course, I will start by discussing minimal submanifolds and scalar/vectorial maximum principles for elliptic and parabolic PDEs. Then, I will use these tools to prove Bernstein type theorems for graphical minimal submanifolds. Finally, I will focus on the mean curvature flow in high codimensions and will demonstrate how to use this powerful method to derive topological results for maps between Riemannian manifolds.
Textbook or/and course webpage:
1. K. Smoczyk, Mean curvature flow in higher codimension: introduction and survey. Springer Proceedings in Mathematics, Vol 7, 231274 (2012). Text also available on arXiv 1104.3222.
2. Y.L. Xin, Minimal submanifolds and related topics, Nankai Tracts in Mathematics, Vol. 16 (2018).
Prerequisites:
Differential Geometry, Riemannian Geometry.
EinsteinMaxwell Manifolds
An Einstein manifold is a (pseudo)Riemannian manifold (M,g) (a spacetime) such that the Ricci tensor is proportional to the metric tensor. Einstein manifolds are the solutions of Einstein's field equations for pure gravity with cosmological constant Λ (Lambda). EinsteinMaxwell manifolds, on the other hand, satisfy EinsteinMaxwell equations consisting of gravity and electromagnetism. These manifolds are not only interesting for physics but also for pure geometry since they are related to many important topics of Riemannian geometry such as Riemannian submersions, homogeneous Riemannian spaces, Riemannian functionals and their critical points, YangMills theory, holonomy groups etc. In these lectures we aim to provide basics of Einstein manifolds and some parts of their classifications. We will also deal with EinsteinMaxwell equations and study on some explicit examples.
Textbook or References:
1. A. L. Besse, “Einstein Manifolds”, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 10. Springer, Berlin (1987).
2. C. LeBrun, “The Einstein–Maxwell equations, Kähler metrics, and Hermitian geometry”, J. Geom. Phys. 91, 163–171 (2015).
Prerequisites:
Basic Differential Geometry (not a must but preferable)
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Contact: ilaydabariss@gmail.com, berkanuze@gmail.com
Activities are supported by Nesin Mathematical Village and Turkish Mathematical Society
